Currently,
R and
S must both be polynomial rings over the same base field.
This function first checks to see whether M will be a finitely generated R-module via F. If not, an error message describing the codimension of M/(vars of S)M is given (this is equal to the dimension of R if and only if M is a finitely generated R-module.
Assuming that it is, the push forward
F_*(M) is computed. This is done by first finding a presentation for
M in terms of a set of elements that generates
M as an
S-module, and then applying the routine
coimage to a map whose target is
M and whose source is a free module over
R.
Example: The Auslander-Buchsbaum formula
Let's illustrate the Auslander-Buchsbaum formula. First construct some rings and make a module of projective dimension 2.
i1 : R4 = ZZ/32003[a..d];
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i2 : R5 = ZZ/32003[a..e];
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i3 : R6 = ZZ/32003[a..f];
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i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
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i5 : pdim M
o5 = 2
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Create ring maps.
i6 : G = map(R6,R5,{a+b+c+d+e+f,b,c,d,e})
o6 = map(R6,R5,{a + b + c + d + e + f, b, c, d, e})
o6 : RingMap R6 <--- R5
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i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 13408a - 7853b - 8408c + 9600d - 14965e, - 15675a - 13036b + 8812c - 1243d + 12841e, 6716a + 7014b + 11792c - 4002d - 5202e, 454a + 12654b + 565c + 10509d + 3020e})
o7 : RingMap R5 <--- R4
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The module M, when thought of as an R5 or R4 module, has the same depth, but since depth M + pdim M = dim ring, the projective dimension will drop to 1, respectively 0, for these two rings.
i8 : P = pushForward(G,M)
o8 = cokernel | c -de |
| d bc-ad+bd+cd+d2+de |
2
o8 : R5-module, quotient of R5
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i9 : pdim P
o9 = 1
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i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0..1, 0}
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i11 : pdim Q
o11 = 0
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Example: generic projection of a homogeneous coordinate ring
We compute the pushforward N of the homogeneous coordinate ring M of the twisted cubic curve in P^3.
i12 : P3 = QQ[a..d];
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i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
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The result is a module with the same codimension, degree and genus as the twisted cubic, but the support is a cubic in the plane, necessarily having one node.
i14 : P2 = QQ[a,b,c];
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i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
9 3 10 3 5 9 2 3 3 3
o15 = map(P3,P2,{-a + b + -c + d, --a + -b + -c + -d, -a + -b + -c + --d})
8 2 7 8 9 8 9 2 4 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 19864979703787146880ab-14399986205801518560b2+11982993206598432000ac-14805698677155033480bc-16881610686591402000c2 11351416973592655360a2-5961861863522157600b2-1835432406708104160ac-3812447163333087720bc-10740869145939974400c2 185048689116281613848033655437240928225696b3-4466584860596399355747134845445102474405280b2c+470116657587772363135317780108952925383104000ac2-299985985653836255722986090110911494426391200bc2-316057774461007959582464925716951464051320000c3 0 |
{1} | -20598577125524081070a+15939876841936188764b+4881554522187985925c -14273026245720574102a+12176667893577212676b-1425506081180895855c 734155266062031097567227543110537629381384116a2-926165446287052138251025245341685223837243752ab+347659450430512387497428461924329694227167292b2-603547379981921378469946958297695079043181220ac+159959507873350810164479074704624502147914780bc-79096446296622133434625828580780199859889775c2 2987154475452480a3-5933779241206320a2b+4147044157279800ab2-1026014644718784b3-3375911257096180a2c+3572517063623760abc-892783945801440b2c+147975856109100ac2+68509306296900bc2+977270200918875c3 |
2
o16 : P2-module, quotient of P2
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i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
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i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
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i19 : ann N
3 2 2
o19 = ideal(2987154475452480a - 5933779241206320a b + 4147044157279800a*b -
-----------------------------------------------------------------------
3 2
1026014644718784b - 3375911257096180a c + 3572517063623760a*b*c -
-----------------------------------------------------------------------
2 2 2
892783945801440b c + 147975856109100a*c + 68509306296900b*c +
-----------------------------------------------------------------------
3
977270200918875c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.